Final answer:
The average rate of change of f(x) = sin x over the interval [5π/6, 3π/2] is calculated as -4.5/π by determining the function values at both endpoints and then dividing the difference by the interval length.
Step-by-step explanation:
The average rate of change of a function over an interval is calculated by finding the difference in function values at the endpoints divided by the length of the interval. For the function f(x) = sin x, we can evaluate the rate of change over the interval [5π/6, 3π/2] by substituting x with the interval endpoints. Therefore:
f(5π/6) = sin(5π/6) = 1/2, and f(3π/2) = sin(3π/2) = -1.
The change in f(x) is (-1) - (1/2) = -3/2.
The change in x is 3π/2 - 5π/6 = π/3.
Thus, average rate of change = (-3/2) / (π/3) = -9/π6 = -4.5/π.